Q. Jesus said, referring to himself, “The Son of Man must suffer many things and be rejected by the elders, the chief priests and the teachers of the law, and he must be killed and on the third day be raised to life.” But Jesus died on a Friday and rose on a Sunday. How was that the “third day”?
The answer has to do with how people in the biblical culture reckoned time. Today was considered the first day, tomorrow the second day, and the day after tomorrow the third day. The day before yesterday was considered the third day going in the other direction. There is a Hebrew idiom that means “it was not like that in the past” that says literally, “It was not like that yesterday, three days,” meaning, “It was not like that yesterday or the day before yesterday.”
We get one clear indication of this usage in the reply Jesus gave when he was told that Herod wanted to kill him. He said, “I will keep on driving out demons and healing people today and tomorrow, and on the third day I will reach my goal. In any case, I must press on today and tomorrow and the next day.” This shows clearly that the “third day” is the day after tomorrow.
So the way people reckoned time, Friday would have been the first day, Saturday the second day, and Sunday the third day.
Good Question 
From a Hebrew perspective Jesus could also have been crucified on Wednesday and rose on Saturday. Interesting perspective I first heard from a Messianic Jew. Makes more sense to me than the stock answer.
That might be one way to approach the phrase “the third day,” but the gospels are unanimous in saying in the resurrection accounts that Jesus rose from the dead on “the first day of the week,” that is, on a Sunday. See Matthew 28:1, Mark 16:2, Luke 24:1, and John 20:1.
The Jews reckoned any part of a day as one day and the next day started at sundown. Jesus was crucified on Friday and his body removed from the cross and buried before sundown (one day). The beginning of the Sabbath (Saturday) started at sundown and ended at sundown (second day). Jesus rose from the dead the following morning (third day).
I have heard this explanation before, but I do not think it is correct. Jesus said to Peter on a Thursday evening, referring to Friday morning, “before the rooster crows today, you will deny three times that you know me.” If the Jews counted partial days as full days, Jesus would have said, “before the rooster crows tomorrow.” Rather, the explanation is as I have said in this post. In this language and culture, today was the first day, tomorrow was the second day, and the day after tomorrow was the third day.